Car rental Bari, Puglia

Definitely Yes on the IDP, because if you happen to be stopped by the police for some reason, you're open to a large fine if you don't have the IDP.

One more question;
renting a GPS from the car rental company seems very expensive at €110/week; is there a better/cheaper alternative?
Anne

In order of simple-to-answer to less-simple...

is it necessary for an American to obtain an International Driving Permit?

Yes. Get it at AAA. Quick, easy, cheap, and legally required, so a no-brainer.

Is it advisable to take out additional insurance even tho my credit
card company gives me coverage?

Probably not. Just be sure that your credit card really does cover what you think it does (get it in writing, and make sure you read all the fine print and understand it all, and you have the amount of coverage you want). Personally, I always use my credit card's coverage and I always decline any additional insurance they offer. With some cards, you MUST do that or else the credit card's coverage is not in effect - one of the little details you must know and understand. I've never had any trouble. Others seem skittish about relying on coverage from a credit card and they just pay more in hopes of avoiding any hassles.

I’ve read that there is an additional 22% fee from the airport. If this is so, would it be cheaper to pick it up in the city?

Can't help you with that, don't know...and a 22% penalty for picking up/returning at an airport sounds...unusual. IME airport locations are sometimes cheaper than in town, but I'm sure there are variances. Check to see if your return day/time is "after hours" at some locations - there may be a big fee for "after hours" services (ie they have to pay someone to go there just for you). You may be able to dodge such fees my adjusting your date.time/location - that may of may not be worth it to you but it's worth checking and trying to find out if that makes a difference. I've learned that early on Sunday mornings are bad times to pick up or drop off a car in small towns (many agencies are closed then) but just up or down the road at the airport they work 24/7 and you have more/cheaper options. You will have to ask questions to find out what's up with that 22% fee.

RE GPS:
Will you be driving solo or have a companion to help navigate? I use Google Maps (I have a free data plan for overseas, but I think you can also just download the maps in advance.

We've used Google Maps on our phones all over Puglia on several trips. No problems at all.

Also, we've always used Europcar all over Italy, including Puglia --- you could see if their rates or fees are any better. I believe we rented once from Foggia after taking the train from Rome, then returned the car to the Bari airport. Have rented and returned from the Brindisi airport, which seemed easier, but I don't remember why.

renting a GPS from the car rental company seems very expensive at €110/week

It sure does.

is there a better/cheaper alternative?

Several. Lots of folks just rely on a phone (this may or may not be problematic, depending on many factors). Personally, I bought a Garmin GPS unit, including "lifetime" map updates for Europe and North America; got it several years ago from Costco for not much more than you would pay for your one-week rental. I've used it everywhere from Ireland to Italy to Latvia to Japan to Mexico and it's a super useful tool (it's also good to be familiar with your GPS before you land in a foreign country...trying to learn/use/understand an agency-supplied, in-dash GPS unit in Lithuanian or Japanese is not easy, believe me, I've tried). I always bring my GPS with updated maps/database from home, it's soooo much easier.

I also have a phone or two with us, an iPad Mini, all with cellular connectivity so I can use any/all of them and (extremely helpful) cross-reference them for conflicting directions (it's quite common that they disagree...I once saved a trip from collapse by cross-referencing as my built-in car GPS in Japan was sending me on a very indirect route which would have caused me to arrive 2 hours late for the only ferry of the day to my destination...my Garmin GPS correctly directed me to a fast, direct highway straight to that ferry, which I made).

Also, always have a good paper map with you, no matter how many gizmos you have and how much you trust them. And have some general idea where you are headed and what route you will take to get there - don't be one of those people that blindly and blithely follows your GPS into a lake...

Thanks everyone for your kind advice.
And @David a big LOL for that last comment!!
Anne

Gosh, David, I thought my husband was the only who did that.

A few years ago, we rented a car from a Bari an in-town office, and returned to the airport. It was no hassle getting out of town, and there was no fee to do the airport return. The fee for picking up at an airport is very common world wide. I have never encountered it for an airport return.
I suggest you look at AutoEurope’s website. It will be very easy to compare prices for in-town vs airport.